I'm a newcomer to Quantum Computing and I'm currently working through a tutorial shown here.
At one point (second last section in Part II), there is an exercise that I'm really struggling with. I'm new to bra-ket notation and, while I can spot some identities given previously in the tutorial that might be useful for the exercise, I'm not able to make much headway.
The question is: Show that for any matrix $M$ and vector $|ψ⟩$, the following identity holds, expressing the length of $M|\psi⟩$:
$\|M|\psi\rangle\|^2 = \langle\psi|M^\dagger M |\psi\rangle.$
My rather poor attempt starts with recognising that $M^\dagger M = MM=M^2=I$, where $I$ is the identity matrix, substituting this into the original formula. This identity was given earlier in the tutorial. This gives me:
$\|M|\psi\rangle\|^2 = M^2\langle\psi|\psi\rangle.$ (1)
Then, noting another identity provided, that $\langle\psi|\psi\rangle = |||\psi\rangle||^2$ and substituting into (1) gives:
$\|M|\psi\rangle\|^2 = M^2|||\psi\rangle||^2$ (2)
At this point I'm basically stuck, I can't tell if I've provided proof, if I'm on the right track but not quite there, or if I'm totally wrong!
The only other thought I'd had is to use the identity:
$\langle\psi|M^\dagger = (M|\psi\rangle)^\dagger$,
but in honesty didn't really get any further than thinking I might need to use it.
I've been able to more or less follow the examples that the tutorial has worked through, but for this exercise I'm struggling even to understand how to proceed. Any advice would be much appreciated.